Problems & Puzzles: Puzzles

 Puzzle 757. Follow-up to Puzzle 756 Emmanuel Vantieghem sent this nice follow-up to Puzzle 756: Find odd numbers  a, b, c  such that the sequence F(n) such that F(1) = a F(2) = b, F(n) = F(n-1) + F(n-2) + c contains no prime at all.         Q. Find your a, b, c values. Observations: 1.- (I omitted the condition  a < b  because it is completely unnecessary: if you find a solution   F(n)  with  a > b, you can take the sequence  F*(n)  defined by  F*(n) = F(n+1) )    2. This problem is somewhat analogue with  problem 31.  Finding a decent covering could solve it, but the fact that all numbers in the sequence must be odd makes it very very difficult.

Contributions came from Claudio Meller, Jan van Delden, Fred Schneider and Emmanuel Vantieghem

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Claudio wrote:

Con a= 8, b=9 y C= 501 se obtienen más de 1184 números no primos

Con 25, 27 y 1541 obtengo 147 términos no primos.

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Jan wrote:

That might be too much to ask for.

Given: G[n+1]=G[n]+G[n-1]+c, with G[0]=a, G[1]=b

Rewrite as: G[n]=A[n]-c with A[n+1]=A[n]+A[n-1], with A[0]=a+c, A[1]=b+c.
Or:              G[n]=F[n-1]*(a+c)+F[n]*(b+c)-c         , with F[-1]=1
Or:              G[n]=F[n-1]*a+F[n]*b+(F[n+1]-1)*c

Where F[n] is de Fibonacci-sequence.

The article A New Fibonacci-like Sequence of Composite (M. Vsemirnov) covers the situation where c=0.
In it the author chooses a set of (r[i],m[i],p[i],c[i]) such that
. the congruences n=r[i] mod m[i] cover the integers.
. a=c[i]F[m[i]-r[i]]  mod p[i] and
. b=c[i]F[m[i]-r[i]+1] mod p[i]
. F[m[i]]=0 mod p[i]
[And (a,b)=1]

We have
A[n]=F[n-1]*a+F[n]*b, substitute formulae for a,b:
A[n] mod p[i] =
c[i]*(F[n-1]*F[m[i]-r[i]]+F[n]*F[m[i]-r[i]+1]) mod p[i]= (*)
c[i]*F[n+m[i]-r[i]] mod p[i]=
c[i]*F[k*m[i]] mod p[i] for some k>=1, since n=r[i] mod m[i] for some i.
A[n]=0 mod p[i], since F[m[i]]|F[k*m[i]] and F[m[i]]=0 mod p[i].
So there is a prime p[i]|A[n] for all n, and since p[i] small compared to A[n], we have A[n] composite.

(*)  F[k+l]=F[k-1]F[l]+F[k]F[l+1] (and from which also the property F[m]|F[n] if and only if m|n follows).

The author’s solution doesn’t have both a,b odd. However his work is an extension of the work of Knuth,
who found (62638280004239857,49463435743205655) [who didn’t use the c[i] above].

We could try and use the same scheme as follows:
Change a to a+c, b to b+c, take c=product(p[i]), p[i] in coverset, since that would preserve p[i]|G[n].
However all coversets discussed in the article contain prime 2, hence c is even, which was not allowed.

One might argue that maybe one could find a coverset not using prime 2.
However I think you might agree that we do need a rule like (*),
because one has to extend composite G[n] from a set [2..N] to all integers somehow.

I think we might have better luck by defining:

H[n]=T[n-1]+T[n-2]+T[n-3], with T[0]=a, T[1]=b, T[2]=c

An extension to trionacci numbers.

In Linear recurrence sequences of composite numbers, chapter 4, the author Jonas Šiurys gives  a solution:
a= 99202581681909167232;
b = 67600144946390082339;
c = 139344212815127987596;
with gcd(a,b,c)=1. However 2 even numbers are used.

However rules similar to (*) are much trickier...

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Fred wrote:

I wasn't able to find a perfect covering set given the time constraints but I did find a set of moduli that guarantees a composite for a minimum of 99.47% and it guarantees an initial sequence of 463 composites (at a minimum).

I constructed a set of moduli which has a cycle of length 10080 and only has 53 values in that cycle which are not divisible by any factors of this number:

961028822148391116866530785 =
3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 41 * 47 * 61 * 71 * 107 * 181 * 211 * 281

The numbers are of the form:
c = 961028822148391116866530785 * n1 + 489436429316952505141575361
a = 961028822148391116866530785 * n2 + 552673760827668088187032942
b = 961028822148391116866530785 * n3 + 201462026909451669613400188

It's just a matter of finding a set of odd composites of this pattern
which are relatively prime to each other.  Due to the size of the numbers, this is a simple step:

For n1 = 0, n2 = 1, n3 = 3, we get:

c = 489436429316952505141575361 = 53 * 307 * 373 * 80644214178147259667
a = 1513702582976059205053563727 = 103 * 9901 * 1484308815502660028509
b = 3084548493354625020212992543 = 509 * 2043409 * 2975849 * 996569582947

Per the factoring, these are relatively prime to each other.

Also the order is significant, flipping a and b does result in different answers.

Here is a screen capture of the moduli set.  The moduli set for "c" is found on row 2, "a" is row 3, "b" is row 4.
For instance, cell S3 means that  "b" must be of the form: 45 mod 47.
Cell G1 (whose formula is in the display window) finds 53 non-zeros out of the 10080.

The interesting thing is that I could find a solution that guaranteed > 94% composites with just 3, 5, 7, 11, 31, 41.

Can anyone find a perfect covering set?

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Emmanuel wrote:

It was a hell of a job, but finally I could find the following  a, b, c :

a = 34645070838898447188004862254466286597623202852789118513550297680601605381
8574266131717600487941201063 (102 digits)
b = 14966963365244319054365268415392264472679965924154087218223476376958335184134779
3731397086724941199065 ( id )
c = 4650952997556328445704238092273930989775254356619718459590888278561977254701832
04031145580541320803285  ( id ).

Every term in the sequence  ( F(n) )  will be divisible by at least one prime of the set
{3, 5, 7, 17, 11, 61, 47, 19, 41, 23, 53, 31, 2207, 107, 2161, 109441, 5779, 2521, 1087, 4481, 103681, 1601, 3041, 2269, 4373, 19441, 181, 3167, 241, 20641, 10749957121, 23725145626561} .

First of all, I looked for odd initial numbers for a Fibonacci-like sequence of composite elements with a covering that uses only odd primes (in problem 31, covering with the prime 2 is admitted).  Then, taking  c  to be the product of the covering primes gives automatically a solution for our problem.

Note that this way of getting  a, b, c  makes it impossible to have relative primality of  a,c  and of   b,c  (a  and  b are relatively prime).

In my opinion, there might be smaller solutions.  However, if one solves the problem the way I did, no spectacular diminishing of the number of digits should be expected.  But, there may be other ways to solve this.

...

Here are two smaller values for  a  and  b ( c  remains the same ) :

a = 149483900952505429036911987254877061234303653849505978343470140460157739831
43074352949592692139622991 (101 digits)
b = 22272586420437209242563238322172756859489578324602952645110151017835819895020
22797016924497336963411 (100 digits).

...

Here is a still better result (with the same  c) :

a = 1477123774381693815397108129878903385941002824052228965359499710130619268725289
063520334348471050217
b = 24996240119054494946454777417299937518305742714812110683693766230306708739251
56149225660116431748763

I believe that there is no smaller result with the same  c.

...

I think it is very difficult to obtain a covering with  c  relatively prime to the primes in the covering set.
The reason is that we can write the elements in our sequence in the form
Fib(n-2)*a+Fib(n-1)*b+(Fib(n)-1)*c  (where  Fib(k)  is the  k-th Fibonacci number).
The sum   Fib(n-2)*a+Fib(n-1)*b  is the  (n-1)-th element in the Fibonacci-like sequence  ( G(n) )  with initial terms  a, b.
Eilas, the regularity with which the primes enter in the Fibonacci sequence (and hence in the sequence  ( G(n) )) is not the same as in the sequence
( Fib(n)-1 )  ( = ( 0, 0, 1, 2, 4, 7, 12, 21, ...) ).

Nevertheless, this remark is intuitive.  I would be happy if I got it wrong !

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